I was wondering what you think about a claim that I sometimes hear in this context – that one of the problems is that universities train too many Ph.D. students. That with a smaller number of math Ph.D. students the above will be less of a problem, and also that this way there will be a smaller number of people dealing with less “serious/important” topics (whatever this means exactly).

This question is certainly relevant to the “adjunct issue”. I heard it before, but always found it somewhat confusing. Specifically to the US, with its market based system, who exactly is supposed to decrease the number of Ph.D.’s? The student themselves should realize how useless in the doctoral degree and stop applying? The individual professors should refuse to accept graduate students? The universities should do this together, in some kind of union? The government? All these questions are a bit different and need untangling.

I was going to write a brief reply, but after Adam asked this question I found a yet another example of lazy journalism by Slate’s “education columnist” Rebecca Schuman who argues:

It is, simply put, irresponsible to accept so many Ph.D. students when you know graduate teaching may well be the only college teaching they ever do.

Of course, Dr. Schuman already has a Ph.D. (from our neighbor UC Irvine) — she just wants others not get one, perhaps to avoid her own fate of an adjunct at University of Missouri. Needless to say, I cannot disagree more. Let me explain.

Universities are not allowed to form a cartel

Let’s deal with the easy part. If the American universities somehow conspired to limit or decrease the number of graduate students they accepts, this would be a classical example of anti-competitive behavior. Simply put, the academia would form a cartel. A textbook example of a cartel is OPEC which openly conspires to increase or decrease oil production in order to control world energy prices. In the US, such activity is against the law due to to the Sherman Act of 1890, and the government/courts have been ruthless in its application (cf. European law to that effect).

One can argue that universities are non-profit institutions and by definition would not derive profit should they conspire, but the law makes no distinction on this, and this paper (co-authored by the celebrity jurist and economist Richard Posner) supports this approach. And to those who think that only giants such as Standard Oil, AT&T or Microsoft have to worry about anti-trust, the government offers plenty of example of going after small time cartels. A notable recent case is Obama’s FTC going after Music Teachers National Association, who have a non-poaching of music students recommendation in their “code of ethics”. Regardless what you think of that case, it is clear that the universities would never try to limit the number of graduate students in a similar manner.

Labor suppy and demand

As legions before her, Schuman laments that pospective grad students do not listen to “reason”:

Expecting wide-eyed, mind-loving intellectuals to embrace the eventual realities of their situations has not worked—yes, they should know better, but if they listened to reason, they wouldn’t be graduate students in the first place. Institutions do know better, so current Ph.D. recruitment is dripping with disingenuousness.

But can you really be “wide-eyed” in the internet era? There is certainly no shortage of articles by both journalists and academics on the “plight” of academic life – she herself links to sites which seem pretty helpful informing prospective graduate students (yes, even the link to Simpsons is helpful). I have my own favorites: this, that, that and even that. But all of these are misleading at best and ridiculous at worst. When I mentioned them on MO, José Figueroa-O’Farrill called them a “parallel universe”, for a good reason.

You see, in this universe people make (mostly) rational decisions, wide-eyed or not. The internet simply destroyed the information gap. Faced with poor future income prospects, graduate students either choose to go elsewhere or demand better conditions at the universities. Faced with a decreasing pool of candidates the universities make an effort to make their programs more attractive, and strive to expand the applicant pool by reaching out to underrepresented groups, foreign students, etc. Eventually the equilibrium is reached and labor supply meets demand, as it always has. Asking the universities (who “do know better”) to have the equilibrium be reached at a lower point is equivalent to asking that Ph.D. programs become less attractive. And I thought Schuman cares…

Impact of government actions

Now, when it comes to distorting of the labor market, the government is omnipotent and with a single bill can decrease the number of graduate students. Let’s say, the Congress tomorrow enacts a law mandating a minimum wage of $60,000 a year for all graduate students. Of course, large universities have small armies of lawyers and accountants who would probably figure out how to artificially hike up the tuition for graduate students and include it in their income, but let’s assume that the law is written to prevent any loopholes. What would happen next?

Obviously, the universities wouldn’t be able to afford that many graduate graduate students. The number of them will plunge. The universities would have to cut back on the TA/recitation/discussion sessions and probably hire more adjuncts to compensate for the loss. In time, this would lower the quality of education or lead to huge tuition increases, or mostly likely a little bit of both. The top private universities who would want to maintain small classes will become completely unaffordable for the middle class. Meanwhile the poorer state universities will commodify their education by creating huge classes with multiple choice machine testing, SAT-style, and further diminishing student-faculty interaction. In fact, to compensate for their increasing cost to universities, graduate students will be asked to do more teaching, thus extending their time-to-degree and decreasing the graduation rates.

Most importantly, this would probably have no positive effect on decreasing competition for tenure track jobs, since the academic market is international. In other words, a decreasing american supply will be immediately compensated with an increasing european supply aided with inflow from emerging markets (ever increasing in quantity and quality production of Ph.D.’s in Asia). In fact, there is plenty of evidence that this would have sharply negative effect on prospects of American students, as decreased competition would result in weaker research work (see below).

In summary, who exactly would be the winners of this government action? I can think of only one group: lazy journalists who would have many new reasons to write columns complaining about the new status quo.

The out of control academics

Let’s go back to Schuman’s “it is [..] irresponsible to accept so many Ph.D. students” quote I mentioned above, and judge in on moral merits. Irresponsible? Really? You are serious? Is it also irresponsible to give so many football scholarships to college students if only a few of them can make it to the NFL? Is it also irresponsible to have so many acting schools given that so few of the students become movie stars? (see this list in my own little town). In the previous post I already explain how graduate schools are apprenticeship programs. Graduate schools give students a chance and an opportunity to succeed. Some students do indeed, while others move to do something else, sometimes succeeding beyond expectations (see e.g. this humorous list).

What’s worse, Schuman implicitly assumes that the Ph.D. study can only be useful if directly applicable to obtain a professorship. This is plainly false. I notice from her CV that she teaches “The World of Kafka” and “Introduction to German Prose”. Excellent classes I am sure, but how exactly the students are supposed to use this knowledge in real life? Start writing in German or become a literary agent? Please excuse me for being facetious – I hope my point is clear.

Does fewer students means better math? (on average)

In effect, this is Adam’s speculation at the end of his question, as he suggested that perhaps fewer mathematics graduate students would decrease the number of “less ‘serious/important’ topics”. Unfortunately, the evidence suggests the opposite. When there is less competition, this is a result of fewer rewards and consequently requires less effort to succeed. As a result, the decrease in the number of math graduate students will lead to less research progress and increase in “less important” work, to use the above language.

To see this clearly, think of sports. Compare this list of Russian Major League baseball players with this list by that of Japanese. What explains the disparity? Are more Japanese men born with a gift to play baseball? The answer is obvious. Baseball is not very popular in Russia. Even the best Russian players cannot compete in the american minor leagues. Things are very different in Japan, where baseball is widely popular, so the talented players make every effort to succeed rather than opt for possibly more popular sport (soccer and hockey in Russian case).

So, what can be done, if anything?

To help graduate students, that is. I feel there is a clear need to have more resources on non-academic options available for graduate student (like this talk or this article). Institutionally, we should make it easier to cross register to other schools within the university and the nearby universities. For example, USC graduate students can take UCLA classes, but I have never seen anyone actually doing that. While at Harvard, I took half a dozen classes at MIT – it was easy to cross register and I got full credit.

I can’t think of anything major the universities can do. Government can do miracles, of course…

P.S. I realize that the wage increase argument has a “fighting straw men” feel. However, other government actions interfering with the market are likely to bring similarly large economic distortions of the academic market, with easily predictable negative consequences. I can think of a few more such unfortunate efforts, but the burden is not on me but on “reformers” to propose what exactly do they want the government to do.

P.P.S. We sincerely wish Rebecca Schuman every success in her search for a tenure track appointment. Perhaps, when she gets such a position, she can write another article with a slightly sunnier outlook.

It’s been awhile since I wanted to rant. Since the last post, really. Well, I was busy. But the time has come to write several posts.

This post is about a number of recent articles lamenting the prevalence of low paid adjuncts in many universities. To sensationalize the matter, comparisons were made with drug cartels and Ponzi schemes. Allegedly, this inequality is causing poverty and even homelessness and death. I imagine reading these articles can be depressing, but it’s all a sham. Knowingly or not, the journalists are perpetuating false stereotypes of what professors really do. These journalists seem to be doing their usual lazy work and preying on reader’s compassion and profound misunderstanding of the matter.

Now, if you are reading this blog, I am assuming you know exactly what professors do (Hint: not just teaching). But if you don’t, start with this outline by my old friend Daniel Liberzon, and proceed to review any or all of these links: one, two, three, four. When you are done, we can begin to answer our main semi-serious question:

What is academia, really, if it’s not a drug cartel or a Ponzi scheme?

I can’t believe this trivial question is difficult to some people, and needs a lengthy answer, but here it is anyway.

Academia rewards industriousness and creativity

This might seem obvious – of course it does! These are the main qualities needed to achieve success doing research. But reading the above news reports it might seem that Ph.D. is like a lottery ticket – the winnings are rare and random. What I am trying to say is that academia can be compared with other professions which involve both qualities. To make a point, take sculpture.

There are thousands of professional sculptors in the United States. The figures vary greatly, but same also holds for the number of mathematicians, so we leave it aside. The average salary of sculptors seems to be within reach from average salary in the US, definitely below that of an average person with bachelor degree. On the other hand, top sculptors are all multimillionaires. For example, recently a sculpture by Jeff Koons sold for $58.4 million. But at least it looked nice. I will never understand the success of Richard Serra, whose work is just dreadful. You can see some of his work at UCLA (picture), or at LACMA (picture). Or take a celebrated and much despised10 million dollar manDale Chihuly, who shows what he calls “art” just about everywhere… But reasonable people can disagree on this, and who am I to judge anyway? My opinion does not matter, nor is that of almost anyone. What’s important, is that some people with expertise value these creative works enough to pay a lot of money for them. These sculptors’ talent is clearly recognized.

Now, should we believe on the basis of the salary disparity that the sculpture is a Ponzi scheme, with top earners basically robbing all the other sculptors of good living? That would be preposterous, of course. Same with most professors. Just because the general public cannot understand and evaluate their research work and creativity, does not mean it’s not there and should not be valued accordingly.

Academia is a large apprenticeship program

Think of graduate students who are traditionally overworked and underpaid. Some make it to graduate with a Ph.D. and eventually become tenured professors. Many, perhaps most, do not. Sounds like a drug cartel to you? Nonsense! This is exactly how apprenticeships works, and how it’s been working for centuries in every guild. In fact, some modern day guilds don’t payanything at all.

Students enter the apprenticeship/graduate program in hopes to learn from the teacher/professor and succeed in their studies. The very best do succeed. For example, this list of Rembrant‘s pupils/assistants reads somewhat similar to this list of Hilbert‘s students. Unsurprisingly, some are world famous, while others are completely forgotten. So it’s not about cheap labor as in drug cartels – this is how apprenticeships normally work.

Academia is a big business

Think of any large corporation. The are many levels of management: low, mid, and top-level. Arguably, all tenured and tenure-track faculty are low level managers, chairs and other department officers (DGS, DUS, etc.) are mid-level, while deans, provosts and presidents/chancellors are top-level managers. In the US, there is also a legal precedent supporting qualifying professors as management (e.g. professors are not allowed to unionize, in contrast with the adjunct faculty). And deservingly so. Professors hire TA’s, graders, adjuncts, support stuff, choose curriculum, responsible for all grades, run research labs, serve as PI’s on federal grants, and elect mid-level management.

So, why many levels? Take UCLA. According to 2012 annual report, we operate on 419 acres, have about 40 thousand students, 30 thousand full time employees (this includes UCLA hospitals), have $4.6 billion in operating revenue (of which tuition is only $580 million), but only about 2 thousand ladder (tenure and tenure-track) faculty. For comparison, a beloved but highly secretiveTrader Joe’s company has about $8 billion in revenue, over 20 thousand employees, and about 370 stores, each with 50+ employees and its own mid and low-level management.

Now that you are conditioned to think of universities as businesses and professors as managers, is it really all that surprising that regular employees like adjuncts get paid much less? This works the same way as for McDonalds store managers, who get paid about 3 times as much as regular employees.

Higher echelons of academia is a research factory with a side teaching business

Note that there is a reason students want to study at research universities rather than at community colleges. It’s because these universities offer many other more advanced classes, research oriented labs, seminars, field works, etc. In fact, research and research oriented teaching is really the main business rather than service teaching.

Think revenue. For example, UCLA derives 50% more revenue from research grants than from tuition. Places like MIT are giving out so many scholarships, they are loosing money on teaching (see this breakdown). American universities cannot quit the undergraduate education, of course, but they are making a rational decision to outsource the low level service teaching to outsiders, who can do the same work but cheaper. For example, I took English in Moscow, ESL at a community college in Brooklyn, French at Harvard, and Hebrew at University of Minnesota. While some instructors were better than others, there was no clear winner as experience was about the same.

So not only the adjunct salaries are low for a reason, keeping them low is critical to avoid hiring more regular faculty and preventing further tuition inflation. The next time you read about adjuncts barely making a living wage, compare this to notorious Bangalore call centers and how much people make over there (between $100 and $250 a month).

Academia is a paradise of equality

College professors are different from drug gangsters not only in the level of violence, but also in a remarkable degree of equality between universities (but not between fields!) Consider for example this table of average full professor salaries at the top universities. The near $200,000 a year may seem high, but note that this is only twice that of average faculty at an average college. Given that most of these top universities are located in the uber-expensive metropolitan areas (NYC, Boston, San Francisco, Los Angeles,etc.), the effect is even further diminished.

Compare this with other professions. Forget the sculptors mentioned above whose pay ratios can go into thousands, let’s take a relatively obscure profession of an opera singer (check how many do you know from this list). Like academia and unlike sculpture, the operas are greatly subsidized by the governments and large corporations. Still, perhaps unsurprisingly, there is a much greater inequality than in academia. While some popular singers like Dmitri Hvorostovskymake over $3 million a year, the average salary is about $100,000 a year, giving a ratio of 30+.

In other words, given that some professors are much better than others when it comes to research (not me!), one can argue that they are being underpaid to subsidize the lackluster efforts of others. No wonder the top academics suffer from the status-income disequilibrium. This is the opposite of the “winner takes all” behavior argued by the journalists in an effort to explain adjuncts’ plight.

Academia is an experience

People come to universities to spend years studying, and they want to enjoy those years. They want to hear famous authors and thinkers, learn basic skills and life changing stories, make lasting friendships, play sports and simply have fun. Sometimes this does not work out, but we are good at what we do (colleges have been perfecting their craft for hundreds of years). Indeed, many students take away with them a unique deeply personal experience. Take my story. While at Moscow University, I heard lectures by Vladimir Arnold, attended Gelfand’s Seminar, and even went to a public lecture by President Roh Tae-woo. It was fun. While at Harvard, I took courses of Raoul Bott and Gian-Carlo Rota (at MIT), audited courses of such non-math luminaries as Stephan Thernstrom and William Mills Todd, III, and went to public lectures by people like Tim Berners-Lee, all unforgettable.

So this is my big NO to those who want to replace tenured faculty with adjuncts, leveling the academic salaries, and commodifying the education. This just would not work; it is akin to calls for abolition of haute cuisine in favor of more fast food. In fact, nobody really wants to do either of these. The inexpensive education is already readily available in the form of community colleges. In fact, students apply in large numbers trying to get to a place like UCLA, which offers a wide range of programs and courses. And it’s definitely not because of our celebrity adjuncts.

In conclusion

Academia is many things to many people. There are many important reasons why the ladder faculty are paid substantially better than TA’s and adjuncts, reasons both substantive and economical. But at no point does the academia resemble the Ponzi schemes and drug cartels, which are famous for creating the economic devastation and inequality (and, um, illegal). If anything, the academia is the opposite, as it creates economic opportunities and evens the playing field. And to those educational reformers who think they know better: remember, we heard it all before…

I was thinking about the Putnam competition and the GRE Mathematics test in the context of graduate admissions. Are they useful? If yes, which one is more relevant? After crunching some numbers, I concluded that while they are useful to some extend, there are problems with both. Even worse, a number of students who fall in the gap between “very good” and “exceptional”, are ill served with either.

1. Graduate admissions in mathematics

As I mention in my earlier post, every year the US produces around 1,600 Ph.D.’s in mathematical sciences (math, applied math, statistics) from over 100 accredited programs, of which about 900 are US citizen and permanent residents. If you restrict to mathematics alone, the numbers drop by about 25% to about 1200. The overall 10 year completion rate is about 50% according to the Council of Graduate Schools study, so perhaps about 3,000-3,200 students start graduate programs.

As a general rule, graduate programs in mathematics explicitly ask for the GRE Subject test scores, but are often happy to hear about the Putnam results as well. In fact, some “how to” guides now recommend taking Putnam exam (and Putnam prep classes!) on par with the GRE test and REU programs (see e.g. here and there). How the schools use either data is probably quite a bit different, and is the other side of our main question.

2. GRE Mathematics Subject test in numbers

The GRE Subject tests are developed and administered by ETS, which is nominally non-profit, but with about 1 billion dollars in revenue. For a quick comparison with a for-profit, non-profit and public institutions, e.g. New York Times Corp, Harvard and UCLA, had 2.3, 3.7 and 4.3 bln dollars in 2011 operating revenues, respectively.

From the official GRE test preparation publication: “The questions are classified approximately as follows: calculus (50%), algebra (25%) and other topics (25%).” This is already unfortunate, but more on that later. Here are these “other topics”:

Introductory real analysis (sequences and series of numbers and functions, continuity, differentiability and integrability, elementary topology of R), discrete mathematics (logic, set theory, combinatorics, graph theory, and algorithms), general topology, geometry, complex variables, probability and statistics, and numerical analysis. The above descriptions of topics covered in the test should not be considered exhaustive […] (emphasis mine – IP)

The GRE Guide gives .92 value for the KR20 reliability test, a solid measure suggesting the test has many questions leading to different scores between strong and weak students. The students have 170 minutes for about 65 questions. The scores are on the scale from 200 to 990, are rounded to nearest multiple of 10, with standard errors 31 points, and 44 for the differences. In other words, if I understand correctly (the guide is vague on this), one should not reliably compare students with scores differing by 50 points of less. I am doubtful most grad schools follow that.

In the same GRE guide, ETS reports that there were about 12,800 test takers in four years (July 2008 to June 2011), roughly 3200 a year. This loosely coincides with our graduate student data, as the students take on average one GRE Subject test. In other words, all students with GRE scores get accepted somewhere. So one should not be surprised to see a high correlation (but not necessarily causation) between grad school ranking and GRE Subject scores. Curiously, ETS’s own study says GRE General are a very poor predictor of success in math graduate programs, at least when it comes to GPA and graduation rate.

So how do grad schools use the GRE Math scores? That’s very much unclear. Of course, all schools gather the statistics like averages of those applied, admitted and/or accepted (reported to the dean, external department reviewers, the NRC study, the US News, etc.), but very few make it publicly available. In a rare moment of openness, Penn State admits what amounts to not much use of GRE scores: their average scores vary widely over the years, swinging from 650 to 890, with a positive trend in recent years. In a general MO discussion on this, Pete Clark writes that University of Georgia does not require GRE Subject, so he looks for high GRE General scores. UCLA is a bit evasive: “those we offer admission to have GRE subject scores in or above the 80th percentile” which according to GRE chart amounts to minimum of about 790, suggesting relevance. MIT is blunt but imprecise: “There is no minimum GRE test score required, but if the score on the math subject GRE is not very high, evidence of excellence must be present elsewhere in the application or in the letters of recommendation.” UPenn is actually helpful: “[GRE Math score] should be at least 750, though applicants with somewhat lower scores may be admitted if the rest of their application is sufficiently strong,” and that the recent average score is 820. This all makes a very foggy picture.

3. Putnam competition in numbers

The premise is simple: first Saturday in December, 6 hours (in two sittings) to solve 12 problems in all areas of mathematics, maximum of 10 points per problem. Joe Gallian wrote a nice summary. The problems are difficult: the maximal score 120 is achieved only very occasionally, once in about 10-15 years. The median score is often either 0, 1 or 2 (out of 120!), and the mean is between 5 and 10 points. I bet it must be depressing to spend 6 hours and get no or almost no points.

The top 5 scorers are “Putnam Fellows”, another 18-20 are “in the money”, and about 50-60 get “honorable mention”. In 2011, there were “4,440 students from 572 colleges and universities in Canada and the United States”. The historical data shows that there is a clear correlation between doing well on Putnam and doing well in mathematics, which is even more pronounced for the top 25, and especially Putnam fellows.

Of course, the competition is not aimed at helping graduate admissions, as emphasized by the mid-March results date (way after the applications are due and the admission decisions are made). It does not even make the scores available in any official format. In fact, historically, it is primarily a team competition, a nerdy alternative to college athletics. Finally, a competition is not necessarily similar to do research. As Kedlaya said, “A contest problem is meant to be solved in the space of minutes or hours, whereas in research, one sometimes works on the same problem for days, months, occasionally even years.”

(a)GRE Math. While useful to some extend, mostly for the middle and bottom scoring students, it is largely useless for most of the better prepared students. Indeed, in the “upper middle range” of 75 to 90 percentile, the test scores range between 770 and 850, comprising about 500 students every year. By the rules of GRE, many of these students cannot be even compared. Those who can, it is unclear whether they really are better candidates for doing research and teaching in mathematics. Indeed, the excessive emphasis on calculus, real analysis and linear algebra shows the student’s ability to memorize concepts and quickly perform routine tasks. This does not test problem solving. Neither do “other topics” which are heavily testing definitions of a group, ring, metric space, etc. I bet the performance in this part strongly correlates with the quality of the undergraduate institution: better colleges offer more serious math classes, and GRE Math preparation classes, which cover these basic topics; others do not.

For the top 10%, the GRE Math scores does distinguish between them, but that’s hardly necessary. Of the top 250-300 students over half of them are international and often come with accolades like “the best student in N years from the XYZ university.” Last year I recall even one European student described as “the best student since World War II from … country”. Those 100-150 that are from the US, are well served with numerous REU programs both national and at their home universities, by the Budapest and Moscow semesters, Putnam, IMO and other competitions, etc. Their GRE scores seem irrelevant in retrospect.

Now, using AMS Classification, Group I of 48 top math graduate programs graduates about 550 Ph.D’s. All are research oriented. I am guesstimating that they must be accepting c. 800 students in total. So after the top 300 are accepted, how are they suppose to choose the next 500 if GRE is irrelevant?

(b) Putnam. Even though a majority receive only single digit score, there is a clear benefit for the top programs to know who the winners are. The top 25 individuals, clearly possess excellent problem solving abilities, which is useful in a number of areas of mathematics. The are multiple problems with this. First, it would be nice to have the list of winners available by December. Second, it would be nice if Putnam is offered overseas. But even for the US/Canada based students, as it stands, the senior’s performance is not counted in admissions due calendar issues. Since students often are encouraged to take their junior year abroad, the best performance they can include in their applications is from their sophomore year, which is often inferior to their senior year performance. So with exception of the truly top students, Putnam results are not used in the admissions.

5. A modest proposal, Russian style

(a) GRE Math. Split the GRE Mathematics into two parts. Keep Calculus/Linear Algebra in the first half, more or less in the same multiple choice form as you have now. It is clearly helpful for middle and bottom tier students and programs. For the second part, make it a no-hard-math-required problem solving style. Make many relatively simple problems, much much simpler than IMO problems, more like Moscow olympiads for the freshman-sophomore HS years (8-9th year out of 11). This would allow relatively unbiased testing of problem solving, extremely useful to mathematics programs. Both scores would need to be reported (kind of like 4 GRE General scores).

As revenue figures suggest, ETS is essentially a large utility company which does not want to rock the boat. But it has made changes before, and this particular change would be relatively painless and have the added advantage that no “other fields” need to be argued about – all students will know exactly what is the scope of the test.

(b) Putnam. Ugh. It’s true that “if it ain’t broke, don’t fix it“, so I don’t want to propose major changes. Just three minor tweaks, which will not change the core competition, but hopefully will make it more democratic and helpful for graduate admissions.

* First, move the competition to late September, so the scores can be revealed before Jan 1. I really don’t see what exactly is hard about that. Perhaps, some Putnamprepclasses will have to be moved to the Spring. So what?

* Second, open it for international students. I know, I know, time difference, language issues, etc. Whatever, keep it on the US time and only in English, as it is now. If the overseas students want to participate, they might have to do this at night perhaps (simply allow unlimited tea, coffee and Red Bull). This is still better than not giving them an opportunity at all. Another issue is trust (in foreign faculty supervisors). For that, use the technology. Reveal the problem on some website for all at once. Videotape what’s happening in all rooms where the competition is taking place. Have all solutions uploaded as .pdf files to the main server within minutes after the end of the competition (they should still be graded locally, with top scores re-graded at a central location). While some of this might be an obstacle for some universities in poor countries, the majority of foreign universities already have all the necessary technology to make this happen.

* Third, and most controversially, at least for the US/Canadian students allow an easy “parallel track”. That is, come up with substantially easier problems which can be administered at the same time in parallel. The students should be given a choice – either real problems which are hard, or easier problems which do not count. This would be good for students’ morale as a means to prevent the annual 40% of 0 scores, and the scores can be useful for admission. I am modelling this based on the widely successful Tournament of Towns, which has two levels and two tracks (harder and easier), see this problem archive.

P.S. Full disclosure: I took GRE Math in 1994 and received maximal score available at that time. I recall finishing early, but missing a couple of problems possibly due to some English language difficulties. I did not participate in the Putnam – was busy in Moscow. More recently, I also participated in graduate admissions, but everywhere above made sure I use only open sources and no “inside information”.

Imagine you have plenty of money and dozens of volunteers. You decide to award one or two fellowships a year to the best of the best of the best in math sciences. Easy, right? Then how do you repeatedly fail at this, without anyone notice? Let me tell you how. It’s an interesting story, so bear with me.

A small warning. Although it may seem I am criticizing Hertz Foundation, my intention is to show its weakness so it can improve.

What is the Hertz Foundation?

Yesterday I wrote a recommendation letter to the Hertz Foundation. Although a Fellow myself, I never particularly cared for the foundation, mostly because it changed so little in my life (I received it only for two out of five years of eligibility). But I became rather curious as to what usually happens to Hertz Fellows. I compiled the data, and found the results quite disheartening. While perhaps excellent in other fields, I came to believe that Hertz does barely a mediocre job awarding fellowships in mathematics. And now that I think about it, this was all completely predictable.

First, a bit of history. John Hertz was the Yellow Cab founder and car rental entrepreneur (thus the namesake company), and he left a lot of money dedicated for education in “applied physical sciences”, now understood to include applied mathematics. What exactly is “applied mathematics” is rathercontentious, so the foundation wisely decided that “it is up to each fellowship applicant to advocate to us his or her specific field of interest as an ‘applied physical science’.”

In practice, according to the website, about 600 applicants in all areas of science and engineering apply for a fellowship. Applications are allowed only either in the senior year of college or 1st year of grad school. The fellowships are generous and include both the stipend and the tuition; between 15 and 20 students are awarded every year. Only US citizen and permanent residents are eligible, and the fellowship can be used only in one of the 47 “tenable schools” (more on this below). The Foundation sorts the applications, and volunteers interview some of them in the first round. In the second round, pretty much only one person interviews all that advanced, and the decision is made. Historically, only one or two fellowships in mathematical sciences are awarded each year (this includes pure math, applied math, and occasionally theoretical CS or statistics).

Forty years of Math Hertz Fellowships in numbers

The Hertz Foundation website has a data on all past fellows. I compiled the data in Hertz-list which spanned 40 years (1971-2010), listed by the year the fellowship ended, which usually but not always coincided with graduation. There were 67 awardees in mathematics, which makes it about 1.7 fellowships a year. The Foundation states that it awarded “over 1000 fellowships” so I guess about 5-6% went into maths (perhaps, fewer in recent years). Here is who gets them.

1)Which schools are awarded? Well, only 44 US graduate programs are allowed to administer the fellowships. The reasons (other than logistical) are unclear to me. Of those programs that are “in”, you have University of Rochester (which nearly lost its graduate program), and UC Santa Cruz (where rumors say a similar move had been considered). Those which are “out” include graduate programs at Brown, UPenn, Rutgers, UNC Chapel Hill, etc. The real distribution is much more skewed, of course. Here is a complete list of awards per institution:

In summary, only 15 universities had at least one award (34%), and just 7 universities were awarded 54 fellowships (i.e. 16% of universities received 81% of all fellowships). There is nothing wrong with this per se, just a variation on the 80-20 rule you might argue. But wait! Hertz Foundation is a non-profit institution and the fellowship itself comes with a “moral commitment“. Even if you need to interfere with “free marketplace” of acceptance decisions (see P.S. below), wouldn’t it be in the spirit of John Hertz’s original goal, to make a special effort to distribute the awards more widely? For example, Simons Foundation is not shy about awarding fellowship to institutions many of which are not even on Hertz’s list.

2) Where are they now? After two hours of googling, I located almost all former fellows and determined their current affiliations (see the Hertz-list). I found that of the 67 fellows:

University mathematicians – 27 (40%)
Of these, work at Hertz eligible universities – 14, or about 21% of the total (excluding 3 overseas)
At least 10 who did not receive a Ph.D. – 15%
At least 13 are in non-academic research – 19% (probably more)
At least 8 in Software Development and Finance – 12% (probably more)

Now, there is certainly nothing wrong with directing corporate research, writing software, selling derivatives, designing museum exhibits, and even playing symphony orchestra or heading real estate company, as some of the awardees do now. Many of these are highly desirable vocations. But really, was this what Hertz had in mind when dedicating the money? In the foundation’s language, “benefit us all” they don’t.

I should mention that the list of Hertz Fellows in Mathematics does include a number of great academic success stories, but that’s not actually surprising. Every US cohort has dozens of excellent mathematicians. But the 60% drop out rate from academia is very unfortunate, only 21% working in “tenable universities” is dismaying, and the 15% drop out rate from graduate programs is simply miserable. Couldn’t they have done better?

A bit of analysis

Every year, US universities award over 1,600 Ph.D.’s in mathematical sciences, of which over a half go to US citizen (more if you include permanent residents, but stats is not easily available). So they are choosing 1.7 out of over 800 eligible students. Ok, because of their “tenable schools” restriction this is probably more like 300-400. Therefore, less than half of one percent of potential applicants are awarded! For comparison, Harvard college acceptance rate is 10 times that.

Let me repeat: in mathematics, Hertz fellows drop out from their Ph.D. programs at a rate of 15%. If you look into the raw 2006 NRC data for graduation rates, you will see that many of the top universities have over 90% graduation rate in the math programs (say, Harvard has over 91%). Does that mean that Harvard on average does a better job selecting 10-15 grad students every year, while Hertz can’t choose one?

Yes, I think it does. And the gap is further considering that Hertz has virtually no competition (NSF Fellowships are less generous in every respect). You see, people at Harvard (or Princeton, MIT, UCLA, etc.) who read graduate applications, know what they are doing. They are professionals who are looking for the most talented mathematicians from a large pool of applicants. They know which letters need to be taken seriously, and which with a grain of salt. They know which undergraduate research experience is solid and which is worthless. They just know how things are done.

Now, a vast majority of Hertz interviewers are themselves former fellows, and thus about 95% of them have no idea about the mathematics research (they just assume it’s no different from the research they are accustomed to). Nor does the one final interviewer, who is an applied physicist. As a result, they are to some extend, flipping coins and rolling dies, in hope things will work out. You can’t really blame them – they simply don’t know how to choose. I still remember my own two interviews. Both interviewers were nice, professional, highly experienced and well intentioned, but looking back I can see that neither had much experience with mathematical research.

You can also see this lack of understanding of mathematics culture is creeping up in other activities of the foundation, such as the thesis prize award (where are mathematicians?), etc. Of course a private foundation can award anyone it pleases, but it seems to me it would do much more good if only some special care is applied.

A modest proposal

There is of course, a radical way to change the review of mathematics applicants – subcontract it to the AMS (or IMA, MSRI, IPAM – all have the required infrastructure). For a modest fee, the AMS will organize a panel of mathematicians who will review and rank the applicants without interviewing them. The panel will be taking into consideration only students’ research potential, not the university prestige, etc. The Hertz people can then interview the top ranked and make a decision at the last stage, but the first round will be by far superior to current methods. Even the NSA trusts AMS, so shouldn’t you?

Hertz might even save some money it currently spends on travel and lodging reimbursements. The 13% operating budget is about average, but there is some room for improvement. Subcontracting will probably lead to an increase in applications, as AMS really knows how to advertise to its members (I bet you currently receive only about 40 mathematics applications, out of a potential 400+ pool). To summarize: really, Hertz Foundation, think about doing that!

P.S. It is not surprising that the 7 top universities get a large number of the fellowships. One might be tempted to assume that clueless interviewers are perhaps somewhat biased towards famous school names in the hope that these schools already made a good decision accepting these applicants, but this is not the whole story. The described bias can only work for the 1st year grad applicants, but for undergraduate applicants a different process seems to hold. Once a graduate school learns that an applicant received Hertz Fellowship (or NSF for that matter), it has every incentive to accept the student, as the tuition and the stipend are paid by the outside sources now.

P.P.S. Of course, mathematicians’ review can also fail. Even the super prestigious AIM Fellowship has at least one recipient who left academia for bigger and better things.